document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197529>Question 1197529</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Taylors have purchased a 290000 house. they made an initial payment of 10000 and secured a mortgage with interest charged at the rate of 6%/year on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized of 30 years, what monthly payment will the Taylor's be required to make? What is their equity after 5 years, 10 years and 20 years? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830835 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830835\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Home value = $290,000 <BR>\n" );
document.write( "Down payment = $10,000 <BR>\n" );
document.write( "Amount loaned = 290,000 - 10,000 = $280,000\r<BR>\n" );
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document.write( "The monthly payment formula is<BR>\n" );
document.write( "M = (L*i)/( 1-(1+i)^(-n) )<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "L = loan amount<BR>\n" );
document.write( "i = interest rate per month, in decimal form <BR>\n" );
document.write( "n = number of months\r<BR>\n" );
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document.write( "In this case:<BR>\n" );
document.write( "L = 280,000<BR>\n" );
document.write( "i = 0.06/12 = 0.005<BR>\n" );
document.write( "n = 360 months (equivalent to 30 years)\r<BR>\n" );
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document.write( "Let's compute the monthly payment<BR>\n" );
document.write( "M = (L*i)/( 1-(1+i)^(-n) )<BR>\n" );
document.write( "M = (280000*0.005)/( 1-(1+0.005)^(-360) )<BR>\n" );
document.write( "M = 1678.74147042771<BR>\n" );
document.write( "M = 1678.74<BR>\n" );
document.write( "You can use a specialized online calculator such as these<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.calculator.net/loan-calculator.html\">https://www.calculator.net/loan-calculator.html</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.bankrate.com/loans/loan-calculator/\">https://www.bankrate.com/loans/loan-calculator/</a><BR>\n" );
document.write( "to confirm the correct monthly payment.\r<BR>\n" );
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document.write( "Alternatively, you can type this command into a spreadsheet<BR>\n" );
document.write( "<font color=blue>=PMT(0.005,360,280000,0,0)</font><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<font color=blue>=PMT(0.005,360,-280000,0,0)</font><BR>\n" );
document.write( "to compute the monthly payment. \r<BR>\n" );
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document.write( "Yet another alternative is to use a TI83 or TI84 calculator.<BR>\n" );
document.write( "Press the button labeled \"APPS\". Select \"Finance\" then select \"TVM Solver\".<BR>\n" );
document.write( "TVM = time value of money<BR>\n" );
document.write( "This will be the set of inputs<BR>\n" );
document.write( "N = 360<BR>\n" );
document.write( "I% = 0.5<BR>\n" );
document.write( "PV = 280000<BR>\n" );
document.write( "After those inputs are entered, move the cursor down to PMT. Then press the key labeled \"Alpha\" and then \"Enter\" and -1678.74147 should show up<BR>\n" );
document.write( "Using PV = -280000 is equally valid. All we care about really is the absolute value of the result.\r<BR>\n" );
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document.write( "Next, I'll use the remaining balance formula mentioned on this page<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.mtgprofessor.com//formulas.htm\">https://www.mtgprofessor.com//formulas.htm</a><BR>\n" );
document.write( "That formula is<BR>\n" );
document.write( "B = L*( (1+i)^n - (1+i)^p )/( (1+i)^n - 1 )<BR>\n" );
document.write( "where the L, i, and n are the same as before. <BR>\n" );
document.write( "I'm using the letter 'i' to replace the letter 'c' mentioned in the link.<BR>\n" );
document.write( "The p refers to the number of months you are into the loan.\r<BR>\n" );
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document.write( "Let's find the remaining balance after 5 years, aka 5*12 = 60 months<BR>\n" );
document.write( "B = L*( (1+i)^n - (1+i)^p )/( (1+i)^n - 1 )<BR>\n" );
document.write( "B = 280000*( (1+0.005)^360 - (1+0.005)^60 )/( (1+0.005)^360 - 1 )<BR>\n" );
document.write( "B = 260552.199103894<BR>\n" );
document.write( "B = 260552.20<BR>\n" );
document.write( "The remaining balance after 5 years is $260,552.20\r<BR>\n" );
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document.write( "Assuming the home value stays at $290,000, then the home equity value is the difference of these items.\r<BR>\n" );
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document.write( "equity = homeValue - remainingBalance<BR>\n" );
document.write( "equity = 290000 - 260552.20<BR>\n" );
document.write( "equity = 29447.80<BR>\n" );
document.write( "This is the amount of money that the home owner can get out of the home after the debts are paid off.\r<BR>\n" );
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document.write( "Now find the remaining balance after 10 years, aka 10*12 = 120 months<BR>\n" );
document.write( "B = L*( (1+i)^n - (1+i)^p )/( (1+i)^n - 1 )<BR>\n" );
document.write( "B = 280000*( (1+0.005)^360 - (1+0.005)^120 )/( (1+0.005)^360 - 1 )<BR>\n" );
document.write( "B = 234320.029898434<BR>\n" );
document.write( "B = 234320.03<BR>\n" );
document.write( "then,<BR>\n" );
document.write( "equity = homeValue - remainingBalance<BR>\n" );
document.write( "equity = 290000 - 234320.03<BR>\n" );
document.write( "equity = 55679.97\r<BR>\n" );
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document.write( "The steps will be the same for 20 years, but this time we use p = 20*12 = 240<BR>\n" );
document.write( "B = remaining balance after p months<BR>\n" );
document.write( "B = L*( (1+i)^n - (1+i)^p )/( (1+i)^n - 1 )<BR>\n" );
document.write( "B = 280000*( (1+0.005)^360 - (1+0.005)^240 )/( (1+0.005)^360 - 1 )<BR>\n" );
document.write( "B = 151210.041484951<BR>\n" );
document.write( "B = 151210.04<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "equity = homeValue - remainingBalance<BR>\n" );
document.write( "equity = 290000 - 151210.04<BR>\n" );
document.write( "equity = 138789.96\r<BR>\n" );
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document.write( "Answers:\r<BR>\n" );
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document.write( "Monthly payment: <font color=red>$1,678.74</font><BR>\n" );
document.write( "Equity after 5 years: <font color=red>$29,447.80</font><BR>\n" );
document.write( "Equity after 10 years: <font color=red>$55,679.97</font><BR>\n" );
document.write( "Equity after 20 years: <font color=red>$138,789.96</font>\r<BR>\n" );
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document.write( "Note: this assumes the home's value stays at $290,000. Realistically of course this home value will fluctuate over time. <BR>\n" );
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